arXiv:1406.1339v2 [math.AG] 2 Apr 2015
by
Claude Sabbah & Jeng-Daw Yu
Abstract. Given a mixed Hodge module N and a meromorphic function f on a com-plex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module N ⊗ Ef, which extends the construction of [ESY15]. We show the strictness of the push-forward filtered D-module through any projective morphism π : X → Y , by using the theory of mixed twistor D-modules of T. Mochizuki. We consider the example of the rescaling of a regular function f, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss-Manin systems of f in terms of the Harder-Narasimhan filtration of the Kontsevich bundles associated with f.
Contents
1. Introduction. . . 2
Part I. Irregular Hodge filtration and twist by Ef. . . 8
2. Exponentially regular holonomic D-modules. . . 8
3. The mixed twistor D-module attached to Ef. . . 11
4. Strictness for exponentially twisted regular holonomic D-modules 13 5. The irregular Hodge filtration. . . 16
Part II. The case of a rescaled meromorphic function. . . 19
6. Kontsevich bundles via D-modules. . . 19
7. The DX×Cv-module E vf(∗H). . . 22
8. The DX×Cu-module E f /u(∗H). . . 36
9. Relation with the irregular Hodge filtration of E(v:u)f(∗H) . . . 42
Appendix. Brieskorn lattices and Hodge filtration. . . 52
References. . . 55
2010 Mathematics Subject Classification. 14F40, 32S35, 32S40.
Key words and phrases. Irregular Hodge filtration, twisted de Rham complex, holonomic D-module, exponential twist, mixed Hodge D-module, mixed twistor D-D-module, Kontsevich bundle, Gauss-Manin system, Laplace transformation.
(C.S.) This research was supported by the grants ANR-08-BLAN-0317-01 and ANR-13-IS01-0001-01 of the Agence nationale de la recherche.
1. Introduction
1.a. The irregular Hodge filtration. The category of mixed Hodge modules on complex manifolds, as constructed by M. Saito [Sai90], is endowed with the standard operations (push-forward by projective morphisms, pull-back by holomorphic maps, duality, etc.). In particular, the structure of the Hodge filtration in this category is well-behaved through these operations. For a meromorphic function f on a complex manifold X, holomorphic on the complement U of a divisor D of X, and for a mixed Hodge module with underlying filtered DX-module (N, F•N), we will define an
“ir-regular Hodge filtration”, which is a filtration on the exponentially twisted holonomic DX-module N ⊗ Ef, where Ef denote the OX-module OX(∗D) equipped with the twisted integrable connection d + df , that we regard as a left holonomic DX-module.
We note that, although N is known to have regular singularities, N ⊗ Ef has irregular
singularities along the components of the divisor D where f takes the value ∞, hence cannot underlie a mixed Hodge module. Therefore, the irregular Hodge filtration we define on N ⊗ Ef, generalizing the definition of Deligne [Del07], and then [Sab10],
[Yu14], [ESY15], cannot be the Hodge filtration of a mixed Hodge module in the sense of [Sai90]. There is an algebraic variant of this setting, where we assume that f is a rational function on a complex smooth variety X.
Remark 1.1. Such a filtration has been constructed in [ESY15] in the following cases: (a) f extends as a morphism X → P1, D is a normal crossing divisor and the
filtered DX-module (N, F•N) is equal to (OX(∗D), F•OX(∗D)), where the filtration is
given by the order of the pole [Del70]. In such a setting, the filtration was denoted F•(E
f(∗H)), where H is the union of the components of D not in f−1(∞);
(b) X = Y × P1, f is the projection to P1 and (N, F
•N) underlies an arbitrary
mixed Hodge module.
Definition 1.2. By a good filtration F• indexed by Q of a DX-module N, we mean a
finite family Fα+•Nof good filtrations
(1) indexed by Z, parametrized by α in a finite
subset A of [0, 1) ∩ Q, such that Fα+pN ⊂ Fβ+qN for all α, β ∈ A and p, q ∈ Z
satisfying α + p 6 β + q.
We can thus regard it as a single increasing filtration indexed by Q, such that Fα+pN/F<α+pN= 0 for any α, p, except for α in a finite set A of [0, 1) ∩ Q.
For each α, the Rees module RFα+•Nis the graded module defined as
P
pFα+pNzp,
where z is a new (Laurent) polynomial variable. Then we set RF•N:=
L
α∈ARFα+•N.
We can then regard a (usual) good filtration indexed by Z as a good filtration indexed by Q.
Theorem 1.3. Let f be a meromorphic function on X, holomorphic on U = X r D,
where D is a divisor in X. For each filtered holonomic DX-module (N, F•N)
un-derlying a mixed Hodge module one can define canonically and functorially a good
F•DX-filtration F
irr
• (N ⊗ E
f) indexed by Q which satisfies the following properties:
(1) Through the canonical isomorphism (N⊗Ef)
|U = N|U, we have F•irr(N⊗E
f) |U =
F•N|U.
(1)As usual, this is understood with respect to the filtration by the order F
•DX, and goodness means that Fα+pN = 0 p ≪ 0locally on X, and grFα+•Nis gr
FD
(2) For each morphism ϕ : (N1, F•N1) → (N2, F•N2) underlying a morphism of
mixed Hodge modules, the corresponding morphism
ϕf : (N1⊗ Ef, F•irr(N1⊗ Ef)) −→ (N2⊗ Ef, F•irr(N2⊗ Ef))
is strictly filtered.
(3) For each α ∈ [0, 1), the push-forward π+(N ⊗ Ef, Fα+irr•(N ⊗ E
f)) by any
pro-jective morphism π : X → Y is strict.
(4) Let π : X → Y be a projective morphism and let h be a meromorphic
func-tion on Y , holomorphic on V = Y r DY for some divisor DY in Y . Assume that
DX := π−1(DY) is a divisor, and set U = π−1(V ) and f = h ◦ π. Then the
co-homology of the filtered complex π+(N ⊗ Ef, F•irr(N ⊗ E
f)), which is strict by (3),
satisfies
Hjπ+RFirr(N ⊗ Ef) = RFirr(Hjπ+N) ⊗ Eh.
(5) In cases 1.1(a) and 1.1(b) above, the filtration F•irr coincides with the filtration
FDel
• constructed in [ESY15].
The proof of the theorem is given in §5.a and relies much on the theory of mixed twistor D-modules of T. Mochizuki [Moc11b]. This theory allows one to simplify and generalize some of the arguments given in [ESY15], by giving a general framework to treat, from the Hodge point of view, irregular D-modules like Ef. By specializing (3)
to the case where Y is a point we obtain:
Corollary 1.4. For (N, F•N) underlying a mixed Hodge module on a smooth
projec-tive variety X, the spectral sequence attached to the hypercohomology of the filtered
de Rham complex Firr
α+•DR(N ⊗ E
f) degenerates at E
1 for each α ∈ [0, 1).
Remark 1.5. The assumption that D := X r U is a divisor is not mandatory, but simplifies the statement. In general, higher cohomology modules supported on X r U may appear for N ⊗ Ef.
1.b. Rescaling a function. The case 1.1(a) is essentially the only case where we can give an explicit expression for Firr
• E
f(∗H) (see [ESY15], according to 1.3(5)). Recall
that we consider a smooth complex projective variety X together with a morphism f : X → P1. We set P
red = f−1(∞) and P = f∗(∞). We also introduce a
supple-mentary divisor H (which could be empty) having no common components with Pred,
and we assume that D := Pred∪ H has normal crossings. We set U = X r D. We will
also consider X as a complex projective manifold equipped with its analytic topology, that we will denote Xan when the context is not clear.
Our main example in this article, that we consider in Part II, is that of the rescaling of the function f : X → P1. The rescaled function with rescaling parameter v is the
function vf : U × Cv→ C, defined by (x, v) 7→ vf (x). This function does not extend
as a morphism to X × P1→ P1.
We consider the projective line P1
vcovered by two charts Cvand Cuwhose
intersec-tion is denoted by C∗
v, and we regard vf as a rational function vf : X × P1v - - → P1.
We are therefore in the situation in the beginning of the previous subsection, with underlying space X := X × P1
v and reduced pole divisor Pred:= (Pred× P1v) ∪ (X × ∞),
where ∞ ∈ P1
v denotes the point u = 0. We will also set P = (P × P1) + (X × {∞}),
H= H × P1
We denote by E(v:u)f(∗H) the O X×P1
v-module OX×P1v(∗D) equipped with the
con-nection d + d(vf ) (on the open set X × Cv) and d + d(f /u) (on the open set X × Cu).
We denote its restriction to the corresponding open subsets by Evf(∗H) and Ef /u(∗H) respectively. According to Theorem 1.3, it is equipped with an irregular Hodge filtra-tion. We make it partly explicit in Theorem 9.1 (only partly, because around u = 0 we only make explicit its restriction to the Brieskorn lattice, see §9.b).
1.c. Variation of the irregular Hodge filtration and the Kontsevich bundles Regarding now v ∈ C∗ as a parameter and considering the push-forward by
q : X × P1
v→ P1v of the rescaling E(v:u)f(∗H), our aim is to describe the variation
with v of the irregular Hodge filtration Firr
• H
k U, (Ω•
U, d + vdf )
considered in [ESY15], and its limiting behaviour when v → 0 or v → ∞.
The irregular Hodge filtration is conveniently computed with the Kontsevich com-plex. Recall that M. Kontsevich has associated to f : X → P1as in §1.b and to k > 0
the subsheaf Ωk
f of ΩkX(log D) consisting of logarithmic k-forms ω such that df ∧ ω
re-mains a logarithmic (k + 1)-form, a condition which only depends on the restriction of ω to a neighbourhood of the reduced pole divisor Pred= f−1(∞). For each α ∈ [0, 1),
let us denote by [αP ] the divisor supported on Predwith multiplicity [αei] on the
com-ponent Pi of P := f∗(∞) with multiplicity ei. One can also define a subsheaf Ωkf(α)
of Ωk
X(log D)([αP ]) by the condition that df ∧ ω is a section of Ω k+1
X (log D)([αP ]), so
that the case α = 0 is that considered by Kontsevich. Clearly, only those α such that αei∈ Z for some i are relevant. If f is the constant map, then Ωkf = ΩkX(log D). One
of the main results of [ESY15], suggested and proved by Kontsevich when P = Pred,
is the equality, for each k, dim Hk U, (Ω• U, d + df ) = X p+q=k dim Hq(X, Ωpf(α)).
More precisely, for each pair (u, v) ∈ C2 and each α ∈ [0, 1) one can form a
com-plex (Ω•
f(α), ud + vdf ) and it is shown that the dimension of the hypercohomology
Hk X, (Ω•
f(α), ud + vdf )
is independent of (u, v) ∈ C2 and α and is equal to the
above value. The irregular Hodge numbers are then defined as
(1.6) hp,qα (f ) = dim H
q
(X, Ωpf(α)).
We have hp,q
α (f ) 6= 0 only if p, q > 0 and p + q 6 2 dim X. (see Remark 8.20(3) for
the mirror symmetry motivations related to the irregular Hodge filtration.) If f is the constant map, we recover the results of Deligne [Del70, Del71]:
dim Hk(U, C) = dim Hk U, (Ω•
U, d) = dim Hk X, (Ω• X(log D), d) = X p+q=k dim Hq X, ΩpX(log D). The Hodge numbers reduce here to hp,q(X, D) = dim Hq X, Ωp
X(log D)
.
Following the suggestion of M. Kontsevich, let us define the Kontsevich
bun-dles Kk(α) on P1 v. We set Kvk(α) := Hk X, (Ω• f(α)[v], d + vdf ) , Kk u(α) := Hk X, (Ω • f(α)[u], ud + df ) . (1.7)
Using the isomorphism C[u, u−1]−→ C[v, v∼ −1] given by u 7→ v−1, we have a natural quasi-isomorphism (1.8) u•: (Ω• f(α)[v, v−1], d + vdf ) ∼ −→ (Ω• f(α)[u, u−1], ud + df )
induced by the multiplication by up on the pth term of the first complex. Since we
know by the above mentioned results that both modules Kk
v (α), Kuk(α) are free over
their respective ring C[v] or C[u], the identification Hk(u• ) : Hk X, (Ω• f(α)[v, v−1], d + vdf ) ≃ Hk X, (Ω• f(α)[u, u−1], ud + df ) allows us to glue these modules as a bundle Kk(α) on P1
v. The E1-degeneration
property can be expressed by the injectivity (1.9) Hk X, σ>p(Ω• f(α)[v], d + vdf ) ֒−→ Hk X, (Ω• f(α)[v], d + vdf ) , Hk X, σ>p(Ω• f(α)[u], ud + df ) ֒−→ Hk X, (Ω• f(α)[u], ud + df ) , where σ>pdenotes the stupid truncation. Since this truncation is compatible with the
gluing u•, this defines a filtration σ>pKk(α). When restricted to C∗
v, this produces
the family Firr,p
α H k U, (Ω• U, d + vdf ) .
We also notice that the pth graded bundle is then isomorphic to OP1(p)h p,k−p α (f ),
so this filtration is the Harder-Narasimhan filtration F•Kk(α) and the
Birkhoff-Grothendieck decomposition of Kk(α) reads
(1.10) Kk(α) ≃ k L p=0 OP1(p)h p,k−p α (f ).
In particular, all slopes of Kk(α) are nonnegative and we have
deg Kk(α) =
k
X
p=0
p · hp,k−pα (f ).
We will show (see Lemma 6.2) that each Kk(α) is naturally equipped with a
meromorphic connection having a simple pole at v = 0 and a double pole at most at v = ∞. It follows from a remark due to Mochizuki (see Remark 6.3) that the Harder-Narasimhan filtration satisfies the Griffiths transversality condition with respect to the connection. This is a concrete description of the variation of the irregular Hodge filtration (Corollary 6.6).
Our main result concerns the limiting behaviour of the variation of the irregular Hodge filtration when v → 0, expressed in this model.
Theorem 1.11.
(1) The meromorphic connection ∇ on Kk(α) has a logarithmic pole at v = 0 and
the eigenvalues of its residue Resv=0∇ belong to [−α, −α + 1) ∩ Q.
(2) On each generalized eigenspace of Resv=0∇ the nilpotent part of the residue
strictly shifts by −1 the filtration naturally induced by the Harder-Narasimhan filtra-tion.
The proof of Theorem 1.11, which is sketched in §6, does not remain however in the realm of Kontsevich bundles. It is obtained through an identification of the Kontsevich bundles with the bundles Hk(α) obtained from the push-forward
D-modules Hk of E(v:u)f(∗H) (see §1.b) by the projection q : X × P1
v → P1v. Recall that Hk := Rkq ∗DRX×P1 v/P1vE (v:u)f(∗H) is a holonomic D P1
It is equipped with its irregular Hodge filtration Firr
• H
k obtained by push-forward,
according to Theorem 1.3(3). We define the bundles Hk(α) by using this filtration,
and the main comparison tools with the Kontsevich bundles K (α) are provided by Theorems 6.4 and 6.5.
1.d. Motivations and open questions
We have already discussed in [ESY15, Introduction] the motivation coming from estimating p-adic eigenvalues of Frobenius (Deligne) and that coming from mirror symmetry (Kontsevich). We list below some more related questions and possible applications for further investigations.
Numerical invariants of mixed twistor D-modules. The theory of mixed twistor
D-modules, as developed by T. Mochizuki [Moc11b], is the convenient framework to treat wild Hodge theory. However, this theory produces very few numerical invariants having a Hodge flavor (like Hodge numbers, degrees of Hodge bundles, etc.). The irregular Hodge filtration, when it does exist, is intended to provide such invariants. Let us emphasize that, contrary to classical Hodge theory, the irregular Hodge filtration is only a by-product of the mixed twistor structure, but is not constitutive of its definition.
Is there a suitable well-behaved category of wild Hodge D-modules with a forgetful functor to the category of mixed twistor D-modules? What about the expected functorial and degeneration properties? The exponentially twisted Hodge modules should give rise to an object in such a category. Moreover, following the definition due to Simpson of systems of Hodge bundles, we can expect that the objects in this suitable category should carry an internal symmetry (a C∗-action in the case of tame
twistor D-modules). A possible approach to this question would be to search for the desired category as the category of integrable mixed twistor D-modules endowed with supplementary structures on the object obtained by rescaling the twistor variable.
Analogies with Hodge theory. Going further in the direction of Hodge theory, one
may wonder whether the irregular Hodge filtration, when it exists, shares similar properties with the usual Hodge filtration on mixed Hodge modules. For example, for a morphism f : X → P1, the D
X-module Ef underlies a pure integrable twistor
D-module (see Proposition 3.3(2)) and is equipped with an irregular Hodge filtration (see Theorem 1.3 with N = (OX, d)). Let π : X → Y be a projective morphism.
According to the decomposition theorem for pure twistor D-modules [Moc11a], the push-forward π+Ef decomposes, together with its twistor structure, into a direct
sum of possibly shifted simple holonomic D-modules. One can wonder whether the analogues of Kollár’s conjectures (proved by M. Saito [Sai91]) hold for the irregular Hodge filtration of Ef.
Also the question of the limiting behaviour, in the sense of Schmid, of the irregular Hodge filtration raises interesting questions. We treat the case of a tame degeneration (the case of Evf when v → 0) in §7, but the case of a non-tame degeneration (like u → 0
in §8) remains unclear in general. We expect that the good behaviour (by definition) of the mixed twistor modules by taking irregular nearby cycles along a holomorphic function should lead to specific limiting properties for the irregular Hodge filtration, when it exists.
Extended motivic-exponential D-modules. Recall that, following [BBD82, 6.2.4], one
defines the notion of a simple regular holonomic D-module of geometric origin on a smooth complex algebraic variety X if it appears as a simple subquotient in a regular holonomic DX-module obtained by using only standard geometric functors
starting from the case where the variety is a point. In particular, such a simple regular holonomic DX-module is a simple summand of a regular holonomic D-module
underlying a polarizable Q-Hodge module of some weight, as defined by M. Saito [Sai88, Sai90]. It therefore underlies a simple complex polarizable Hodge module. In other words, there exists an irreducible algebraic closed subvariety Z ⊂ X, a Zariski smooth open set Z◦⊂ Z, and an irreducible local system on Z◦, underlying a
polarizable complex variation of Hodge structure (see [Del87]), such that this regular holonomic DX-module corresponds, via the inverse Riemann-Hilbert correspondence,
to the intermediate extension of this local system by the inclusion Z◦ ֒→ X. In
particular, it comes equipped with a good filtration (that induced by the polarizable Q-Hodge module) and the corresponding filtered D-module is a direct summand of the filtered D-module underlying the polarizable Q-Hodge module.
M. Kontsevich [Kon09] has defined the category of motivic-exponential D-modules by adding the twist by Ef for any rational function f to the standard permissible
operations on regular holonomic D-modules of geometric origin on algebraic varieties. By [Moc11b], any such motivic-exponential D-module underlies a pure wild twistor D-module (see [Moc11a]).
There is also the category of extended motivic-exponential D-modules, by autho-rizing extensions of such objects, but we will not consider it here.
One can expect that any motivic-exponential D-module on a complex algebraic variety is endowed with a canonical irregular Hodge filtration, and that this filtration has a good behaviour with respect to the various permissible functors (the six op-erations of Grothendieck, the nearby and vanishing cycles along a function, and the twist by some Ef). Theorem 1.3 is a step toward this expected result.
Remark 1.12 (Hodge filtration in presence of very irregular singularities)
The holonomic DY-modules one obtains as Hkπ+(N ⊗ Ef) when π is any
pro-jective morphism may have irregular singularities much more complicated than an exponential twist of a regular singularity. For example, if Y is a disc, it is shown in [Rou07] that any formal meromorphic connection at 0 ∈ Y can be produced as the formalization at the origin of a connection obtained by the procedure of Th. 1.3(3) for some suitable N on X = Y × P1.
However, these DY-modules come equipped with a good filtration F•H
kπ
+(N⊗Ef)
obtained by pushing-forward Firr
• (N ⊗ E
f). If Y is projective and if for example
Hkπ+(N ⊗ Ef) = 0 except for k = ko then, according to Corollary 1.4, we obtain the degeneration at E1 of the spectral sequence attached to the hypercohomology
of the filtered de Rham complex F•DR H
koπ
+(N ⊗ Ef). Examples of this kind
can be obtained by the procedure of [Rou07] with arbitrary complicated irregular singularities.
Acknowledgments. We thank Maxim Kontsevich for suggesting us the properties
stated in Theorem 1.11 and Takuro Mochizuki for explaining us some of his results on mixed twistor D-modules and his useful comments. In particular, he suggested
various improvements and simplifications to the first version of this article. We owe him the statement of Theorem 6.4. Last but not least, we thank Hélène Esnault for the many discussions we had together and for many suggestions and questions on the subject of this article.
PART I. IRREGULAR HODGE FILTRATION AND TWIST BY EF
2. Exponentially regular holonomic D -modules
2.a. The graph construction. We refer to the expository book [HTT08] or the expository article [Meb04] for basic properties of regular holonomic D-modules.
Let X be a complex manifold and let Pred be a reduced divisor in X. We set
U = X r Pred. Let f be a meromorphic function on X which is holomorphic on U
whose pole divisor P is exactly supported by Pred, i.e., f takes the value ∞ generically
on each irreducible component of Pred. By definition, locally analytically on Pred,
the function f can be written as the quotient of two holomorphic functions with no common factor, such that the zero divisor may intersect Pred in codimension two
in X at most. There exists a proper modification π : X′ → X with X′ smooth,
which is an isomorphism over U , and a holomorphic map f′ : X′ → P1
t, such that
f|π′ −1(U) = f ◦ π|π−1(U). The pole divisor P′ of f′ satisfies Pred′ ⊂ π−1(Pred) =: D′,
and the inclusion may be strict. Let if : U ֒→ U × Ct denote the graph inclusion
of f . The closure Uf of Uf := if(U ) in X × P1t is a closed analytic set of codimension
one, equal to the projection by the proper modification π × Id : X′× P1
t → X × P1t
of the graph if′(X′). The projection p : X × P1t → X induces a proper modification
Uf → X, and the pull-back of U in Uf maps isomorphically to U . In particular, we
have (X × ∞) ∩ Uf ⊂ (Pred× P1t) ∩ Uf. We summarize this in the following diagram.
(2.1) X′ f′ --, if ′ %% π ∼ //i f′(X′) π × Id //X′× P1 π × Id q′ X Uf //X × P1 q ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ ❀ U f 55 r if :: ? OO ∼ //U f ? OO //U × C? OO P1
Let N be a holonomic DX-module. We assume that N is equal to its localization
N(∗Pred) (if not, replace N with N(∗Pred), which is also a holonomic DX-module, by a theorem of Kashiwara). The localized pull-back N′ := π+N(∗D′) consists of a single
holonomic DX′-module. We then recover N as the push-forward π+N′ = H0π+N′
Let us set M′= i
f′,+N′. Then M′= M′(∗(D′×P1t)) and since Supp M′∩(X′×∞) ⊂
(D′× P1
t), we also have M′ = M′(∗[(D′ × P1t) ∪ (X′× ∞)]). We clearly have N′ =
p′+M′ = H0p′+M′.
We set M = (π × Id)+M′= H0(π × Id)+M′. Then
M= M(∗(P × P1t)) = M(∗[(P × P1t) ∪ (X × ∞)]),
and N = p+M = H0p+M. We notice that M does not depend on the choice of
π : X′→ X. We will use the notation M = i
f,⊕N, for which we still have p+if,⊕= Id,
and which coincides with if,+Nif f extends from X to P1(i.e., if we can take π = Id,
so that f′= f ).
Lemma 2.2. If N is regular holonomic, so is M = if,⊕N.
Proof. Indeed, N′ is then regular, hence M′ also, and then M too.
Remark 2.3 (The graph construction for mixed Hodge modules)
Let us now start with a filtered DX-module (N, F•N) underlying a mixed Hodge
module [Sai90]. We still assume that N = N(∗Pred) (if this is not the case, we use the
localization functor in the category of mixed Hodge modules to fulfill the assumption). The construction of §2.a can be done for mixed Hodge modules, by using the corre-sponding functors in the category of mixed Hodge modules. We therefore get a mixed Hodge module (M, F•M) on X × P 1 t such that p+(M, F•M) = H 0p +(M, F•M) = (N, F•N). If f extends as a morphism X → P 1, then (M, F •M) = if,+(N, F•N).
2.b. Exponential twist of holonomic D -modules. The differential df of the function f : U → Ct extends as a meromorphic 1-form on X with poles along Pred.
We denote by Ef the free O
X(∗Pred)-module of rank one equipped with the connection
d + df . For N as in §2.a (in particular, N = N(∗Pred)), we consider the holonomic
DX-module N ⊗O
X E
f.
Lemma 2.4. For M = if,⊕N, we have M ⊗ Et≃ if,⊕(N ⊗ Ef).
This implies N ⊗ Ef ≃ p
+(M ⊗ Et) = H0p+(M ⊗ Et).
Proof. Assume first that f extends as a map X → P1
t. We will work in the chart
centered at ∞ in P1, with coordinate t′, and we will set g = (t′ ◦ f )−1, so that
f−1(∞) = g−1(0). We denote by e1/g the generator of E1/g. We have
ig,+(N ⊗ E1/g) =L k
(N ⊗ E1/g) ⊗ ∂tk′δ(t
′− g)
with its standard DX×Ct′-module structure. There exists thus a unique OX[∂t′]-linear
isomorphism if,+(N ⊗ E1/g)−→ M ⊗ E∼ 1/t
′
induced by
(n ⊗ e1/g) ⊗ δ(t′− g) 7−→ (n ⊗ δ(t′− g)) ⊗ e1/t′. In other words, for each k,
(n ⊗ e1/g) ⊗ ∂tk′δ(t′− g) 7−→ ∂tk′
(n ⊗ δ(t′− g)) ⊗ e1/t′.
By using the same argument as in the proof of [ESY15, (1.6.5)], one shows that this isomorphism is DX×P1-linear.
Let us now consider the general case. By definition,
One then checks that
π+(N ⊗ Ef)(∗D′) = (π+N)(∗D′) ⊗ Ef′ = N′⊗ Ef′,
so if′,+π+(N ⊗ Ef)(∗D′) = M′⊗ Et by the argument above. Then, because Et=
(π × Id)+Et, we have (π × Id)
+(M′⊗ Et) = M ⊗ Et.
2.c. Exponentially regular holonomic D -modules
Lemma 2.5. Assume that M is any regular holonomic DX×P1-module. Then the
push-forward p+(M ⊗ Et) has holonomic cohomology and satisfies Hkp+(M ⊗ Et) = 0 for
k 6= 0.
Proof. The first statement follows from the holonomicity of M ⊗ Et. We can assume
that M = M(∗∞). Let us set M = p∗M. Then M is a regular holonomic DX[t]h∂t
i-module, and p+(M ⊗ Et) is the complex
0 −→ M −−−−−−→ M∂t+ 1
•
−→ 0,
where the•indicates the term in degree zero. Set K = H−1p+(M⊗Et) = ker(∂t+1).
It is DX-holonomic, and the DX-linear inclusion K ֒→ M extends as a natural
DX[t]h∂ti-linear morphism K[t] ⊗ E−t → M . It is clear that K[t] ⊗ E−t is purely irregular along t = ∞ (this is easily seen on the generic part of the support of K) hence, since M is regular, this image is zero, so K = 0.
Definition 2.6. We say that a holonomic DX-module Nexp is exponentially regular if
there exists a regular holonomic DX×P1-module M such that Nexp≃ H0p+(M ⊗ Et).
Proposition 2.7.
(1) If f is meromorphic on X and holomorphic on U = X r D, and if N = N(∗D)
is a regular holonomic DX-module, then N ⊗ Ef is exponentially regular.
(2) Let π : X → Y be a proper morphism and let Nexp be exponentially regular
on X. Then for each j, Hjπ
+Nexp is exponentially regular on Y .
Proof. The first point follows from Lemma 2.2 and Lemma 2.4. For the second point,
set Nexp = H0p+(M ⊗ Et) with M regular on X × P1. We have, according to
Lemma 2.5,
Hjπ+Nexp= Hjπ+(H0pX,+(M ⊗ Et)) = Hj(π+pX,+(M ⊗ Et)) = Hj(p
Y,+(π × Id)+(M ⊗ Et)).
Now, (π × Id)+(M ⊗ Et) = (π+M) ⊗ Et, with π+Mhaving regular holonomic
coho-mology. We thus have Hkp
Y,+Hj(π × Id)+(M ⊗ Et) = 0 for k 6= 0 according to
Lemma 2.5, hence
Hj pY,+(π × Id)+(M ⊗ Et)= H0pY,+Hj(π × Id)+(M ⊗ Et) = H0pY,+ (Hjπ+M) ⊗ Et
.
3. The mixed twistor D -module attached to Ef
If f is a rational function on X with pole divisor P , the twist of a holonomic DX-module by Ef consists first in localizing this module along Pred and then in adding df to its connection. The main property used is that the localization functor on holonomic DX-modules preserves coherence (hence holonomy).
For a filtered holonomic DX-module, the stupid localization functor (∗Pred), which
consists in localizing both the module and its filtration, does not preserve coherence since the localization of a coherent OX-module does not remain OX-coherent. In the
theory of mixed Hodge modules, there is a localization functor which extends the one at the level of regular holonomic D-modules. We will now consider the case of RX-modules and mixed twistor D-modules, in order to treat the Laplace transform
of mixed Hodge modules.
We keep the analytic setting of §2.a. Recall the following notation used in the theory of twistor D-modules (see [Sab05, Moc07, Moc11a]). For a complex mani-fold X, we denote by X the product X ×Czof X with the complex line having
coordi-nate z. The ring RX is the analytification of the Rees ring RFDX :=Lk∈NFkDXzk
attached to the ring of differential operators equipped with its standard filtration by the order. It is locally expressed as OXhðx1, . . . , ðxni, where ðxi := z∂xi.
The smooth case. We denote by EUf /z the RU-module OU equipped with the
z-connection zd + df . By using the same argument as in [Sab04, §2.2], one checks that EUf /z underlies a smooth twistor D-module; equivalently, it corresponds to a harmonic metric on the flat bundle (OU, d + df ). It follows that EUf /z underlies
a polarized variation of smooth twistor structure of weight 0, equivalently a pure polarized smooth twistor D-module.
The stupid localization. Similarly, writing for short OX(∗Pred) := OX ∗(Pred× Cz),
we consider OX(∗Pred) · ef /z := (OX(∗Pred), zd + df ), where we denote the global
section 1 of OX(∗Pred) by ef /z. This is a coherent RX(∗Pred)-module (however, it is
not necessarily RX-coherent). Note also that there is a natural action of z2∂z, by
setting z2∂
z(ef /z) := −f · ef /z in OX(∗Pred). This action commutes with that of the
z-connection. We say that (OX(∗Pred), zd + df ) is integrable (see [Sab05, Chap. 7]). Lemma 3.1. Assume that f : U → C extends as a holomorphic map f : X → P1. Then
OX(∗Pred) · ef /z is RX-coherent.
Proof. The question is local near a point of Pred and, up to shrinking X, we
may assume that f = 1/g for some holomorphic function g : X → C. Then OX(∗Pred) · ef /z= OX(∗{g = 0})e1/gz. If Predhas normal crossings, we choose local
coordinates such that g = xe
, and the relation xe
ðxie 1/xe z = (−e i/xi)e1/x e z gives
the coherence. If Pred is arbitrary, let π : X′ → X be a projective modification over
a neighbourhood of the point of Pred we consider, such that π−1(Pred) has normal
crossings. Set g′= g ◦ π. Then
π+ OX′(∗{g′= 0})e1/g ′z
= H0π+ OX′(∗{g′= 0})e1/g ′z
= OX(∗{g = 0})e1/gz,
since it can be seen that g is invertible on H0π
+ OX′(∗{g′ = 0})e1/g ′z
. By the properness of π, OX(∗{g = 0})e1/gz is then RX-coherent.
Proposition 3.2. If g : X → C is holomorphic, OX(∗{g = 0})e1/gz underlies a pure
wild twistor D-module of weight zero.
As a consequence, the same property holds for OX(∗Pred) · ef /z if f : U → C
extends as f : X → P1.
Proof of Proposition 3.2. This is essentially obvious from the theory of T. Mochizuki
[Moc11a], but we will make the argument precise. Firstly, one can reduce to the case where g = 0 has normal crossings, since pure wild twistor D-module of weight zero are stable by H0π
+, if π is a projective morphism. Here we take π as in the proof of
Lemma 3.1.
Set now U = {g 6= 0} ⊂ X. Let (C∞
U , ∂, h) be the trivial bundle with its standard
holomorphic structure, equipped with its standard metric for which h(1, 1) = 1. Con-sider it as a harmonic Higgs bundle on U with holomorphic Higgs field θ = d(1/g). Since g is a monomial (in local coordinates), this produces a non-ramified good wild harmonic bundle on X, in the sense of [Moc11a, Def. 7.1.7].
For a fixed z (denoted by λ in loc. cit.), denote by Ez the holomorphic bundle
(C∞
U , ∂ + zd(1/g)). The extension PEz defined in [Moc11a, Not. 7.4.1] is nothing
but OX· exp(z/g − z/g). Together with its natural connection, it is isomorphic to
E(1+|z|2)/gz (see Example 7.4.1.2 in loc. cit.). Since there is no Stokes phenomenon
in rank one, the construction QEz of §11.1 in loc. cit. consists only in dividing the
irregular value by 1 + |z|2, so QEz≃ E1/gz (first point of Th. 11.1.2 in loc. cit.). Now,
E1/gz is the canonical prolongation of (C∞
U , ∂, d(1/g), h) as a coherent RX-module.
It is also equal to the RX-module denoted by E in loc. cit. (see §12.3.2). Then one
concludes by using Prop. 19.2.1 of loc. cit.
The twistor localization. Let H be a divisor in X, locally defined by a
holomor-phic function h and let N be a coherent RX(∗H)-module. According to [Moc11b,
Def. 3.3.1], one says that N is twistor-specializable along H if there exists a coher-ent RX-submodule N [∗H] ⊂ N such that, considering locally the graph inclusion
ih: X ֒→ Y := X × C with the coordinate t on C,
• the coherent RY(∗{t = 0})-module ih,+N is strictly specializable along t = 0,
in the sense of [Sab05, §3.4.a],
• ih,+(N [∗H]) is equal to the coherent RY-submodule of ih,+N generated by
the Vt
1 term of the V -filtration (with the convention taken in this article), denoted by
(ih,+N )[∗t].
If N [∗H] exists locally, it is unique, hence exists globally. The category MTMint is introduced in §7.2 of [Moc11b], and the results of loc. cit. imply the following.
Proposition 3.3. Let f be any meromorphic function on X with pole divisor P .
(1) The coherent RX(∗Pred)-module OX(∗Pred) · ef /z is twistor-specializable
along Pred and defines OX(∗Pred) · ef /z[∗Pred] =: EXf /z.
(2) Moreover, EXf /z underlies an object of MTMint(X) extending the object of MTMint(U ) that EUf /z underlies.
(3) If f extends as a morphism f : X → P1, then Ef /z
X = OX(∗Pred) · ef /z and the
(4) Let H be any divisor in X. Then EXf /z(∗H) is twistor-specializable along H
and the corresponding object EXf /z[∗H] underlies an object of MTMint(X).
Proof. Let us start by (3). Let g be a local equation of Pred. Then
ig,+ OX(∗Pred) · e1/gz= ig,+OX(∗Pred)⊗ e1/t
′z
and [Sab09, Prop. 2.2.5] shows that the Vt′
-filtration is constant. Therefore, ig,+ OX(∗Pred) · e1/gz[∗t′] = ig,+ OX(∗Pred) · e1/gz
and thus OX(∗Pred) · ef /z[∗Pred] = OX(∗Pred) · ef /z, as wanted. The remaining
assertion in (3) is then given by Proposition 3.2.
Let us prove (1) and (2). If f does not extend as a morphism X → P1, let
π : X′ → X be as in (2.1). Set D′ = Pred′ ∪ H′. Then, according to [Moc11b,
Prop. 11.2.1], EXf′′/z[∗H′] underlies an object of MTM
int(X′). According to [Moc11b,
Prop. 11.2.6], its push-forward H0π +Ef ′/z X′ [∗H′] underlies an object of MTM int(X). We also have EXf′′/z[∗H ′] = (O X′(∗D′) · ef ′/z
)[∗D′] and we can apply [Moc11b,
Lem. 3.3.17] (because we work with objects of MTM(X′)) to deduce that
H0π+Ef′/z X′ [∗H ′] = H0π +(OX′(∗D′) · ef ′/z )[∗Pred].
On the other hand, we have H0π
+(OX′(∗D′) · ef ′/z
) = OX(∗Pred) · ef /z. Therefore
the latter RX(∗Pred)-module is twistor-specializable along Predand we have EXf /z=
H0π+Ef′/z
X′ [∗H
′]. This concludes (1) and (2).
Lastly, (4) follows from [Moc11b, Prop. 11.2.1].
The Laplace twist. Let f : U → C be as above and let τ be a new variable. We now
consider the function τ f : U × Cτ → C as a meromorphic function on X × Cτ.
Propo-sition 3.3 implies that EX×Cτ f /zτ exists and underlies an object of MTMint(X × Cτ).
In §9 we will also have to consider another variable v and the object EXτ vf /z of MTMint(X × Cv× Cτ).
Proposition 3.4. If f : U → C extends as a morphism f : X → P1, then Eτ f /z X×Cτ =
OX×Cτ(∗Pred) · e
τ f /z.
Proof. The question is local near Pred and, using the notation as above, we have to
prove that EX×Cτ /gzτ = OX×Cτ(∗Pred) · e
τ /gz. Equivalently, we should prove that the
Vt′
-filtration of (ig,+OX×Cτ)(∗{t
′= 0}) · eτ /t′z
is constant. This is obtained through the equation δ(t′− g) ⊗ eτ /t′z
= t′ð
τδ(t′− g) ⊗ eτ /t
′z
.
4. Strictness for exponentially twisted regular holonomic D -modules We will first prove a particular case of Theorem 1.3. Let p : X × P1→ X denote
the projection and let t be the coordinate on the affine line C = P1r{∞}. Recall
that, for (M, F•M) underlying a mixed Hodge module on X × P
1, we have constructed
in [ESY15, §3.1] a filtration FDel
• (M ⊗ E
t) indexed by Q (see Definition 1.2 for the
corresponding Rees construction).
Theorem 4.1. For (M, F•M) underlying a mixed Hodge module, the complex
p+RFDel • (M ⊗ E
In the case where X is a point, this is the statement of [Sab10, Th. 6.1]. If (M, F•M) = if,+(N, F•N) for some morphism f : X → P
1 and some (N, F
•N)
un-derlying a mixed Hodge module on X, one can adapt the proof given in [ESY15, Prop. 1.6.9] for N = OX(∗D), where D is a normal crossing divisor, and f−1(∞) ⊂ D,
but this case is not enough for our purposes. The proof that we give below uses the full strength of the theory of mixed twistor D-modules of T. Mochizuki [Moc11b].
Proof of Theorem 4.1. We first note that the second assertion in the theorem (i.e.,
the vanishing of Hj for j 6= 0) follows from the strictness assertion together with
Lemma 2.5. So let us consider the strictness assertion.
We refer to [ESY15, §§2 & 3] for the notation and results we use here. Given the filtered DX×P1-module (M, F•M) underlying a mixed Hodge module, we associate to
it the Rees module M := RFM=LpFpM· zp, which is a graded RFDX×P1-module.
Its analytification Man (with respect to the z-variable) is part of the data defining
an integrable mixed twistor DX×P1-module, according to [Moc11b, Prop. 12.5.4].
Let us consider the graded RFDX×P1-module RFDel(M ⊗ Et). Our aim is to
prove the strictness (i.e., the absence of z-torsion) of the push-forward modules Hjp
+RFDel(M ⊗ Et). Forgetting the grading, RFDel(M ⊗ Et) can be obtained by
us-ing an explicit expression of the V -filtration as in [ESY15, Prop. 3.1.2]. It is enough to check the strictness property on the corresponding analytic object, by flatness. Now, the analytification RFDel(M ⊗ Et)
an
can be obtained by using the analytic V -filtration, by making analytic the formula of [ESY15, Prop. 3.1.2]. We then use that the V -filtration behaves well by push-forward for mixed twistor D-modules, ac-cording to results of [Moc11b]. This is the main argument for proving Theorem 4.1.
Let us denote by fM the (stupidly) localized module M (∗∞) and byFM
the (not graded) (RFDX×P1)[τ ]hðτi-module Mf[τ ] ⊗ Etτ /z. By Proposition 3.4, this is also
M[τ ] ⊗ Etτ /z. Similarly, (FM
)an denotes its analytification with respect to both τ
and z. We can use Proposition 3.3 together with [Moc11b, Prop. 11.3.4] to ensure that (FM)anunderlies an integrable mixed twistor D-module.
Let p : X × P1× C
τ → X × Cτ denote the projection. Then p+FMan is strict,
each Hjp
+FMan is strictly specializable along τ = 0 and the Vτ-filtration satisfies
Vτ
• H
jp
+FMan = Hjp+(V•τ
FMan). Indeed, these properties are satisfied according
to the main results of [Moc11b].
We will now adapt the proof given in [ESY15, §3.2], which needs a supplementary argument, since we cannot argue with (3.2.2) in loc. cit.
According to [ESY15, Prop. 3.1.2], we have a long exact sequence · · · −→ Hjp+Vατ
FMan−−−−−→ Hτ − z jp +Vατ
FMan−→ Hjp
+(RFDel(M ⊗ Et))an−→ · · ·
that we can thus rewrite as
(4.2) · · · −→ VατH jp+FMan−−−−−→ Vτ − z ατHjp+FMan
−→ Hjp+(RFDel(M ⊗ Et))an−→ · · ·
Let us first check that τ − z is injective on each Hjp
+FMan. In the case considered
in [ESY15, §3.2], we could use (3.2.2) of loc. cit., and when X is reduced to a point, the argument in [Sab10] uses the solution to a Birkhoff problem given by M. Saito. We do not know how to extend the argument of [Sab10] to the case dim X > 1.
Lemma 4.3. Let Y be a complex manifold, N an an R
Y-module which underlies a
mixed twistor D-module in the sense of [Moc11b]. Let h be a holomorphic function
on Y . Then the action of h − z is injective on N an.
Proof. Since a mixed twistor D-module is in particular an object of the category
MTW(Y ) (see [Moc11b, §§7.1.1 & 7.2.1]), a simple extension argument with respect to the weight filtration allows us to reduce to the case where Nan underlies a pure
wild twistor D-module (as defined in [Moc11a]). Since the question is local on Y , we fix some yo∈ Y and work locally near yo.
Assume first that Nan underlies a smooth pure twistor D-module. Then it is a
locally free OY-module with z-connection, and the injectivity of h − z is clear.
In general, we know that N an has a decomposition by the strict support
(see [Sab05, §3.5], [Moc11a, §22.3.4], [Sab09, §1.4]) and we can therefore assume that, near yo, Nan has strict support a germ of an irreducible closed analytic subset
Z ⊂ Y at yo. On a dense open set Zo of the smooth part of Z, due by Kashiwara’s
equivalence for pure twistor D-modules (see loc. cit.), we are reduced to the smooth case considered above and the injectivity holds. Therefore, ker[(h− z) : Nan→ Nan]
is supported on a proper closed analytic subset Z′ of Z in the neighbourhood of y o.
Let F•N
an be a good filtration of Nanas an (R
FDY)an-module (which exists since
we work locally on Y ). Then for each k, ker[(h − z) : FkNan → FkN an] is a
coherent OY ×Cz-submodule of N
an supported on Z′. The (R
FDY)an-submodule
that it generates is a coherent (RFDY)an-submodule of Nan supported on Z′. It is
therefore zero since Nan has strict support equal to Z. Since this holds for any k,
we conclude that ker[(h − z) : Nan→ Nan] = 0.
Since Hjp
+FMan underlies a mixed twistor D-module, we infer from Lemma
4.3 that τ − z is injective on each Hjp
+FMan. We conclude that the long exact
sequence (4.2) splits into short exact sequences and therefore Hjp
+(RFDelM)an is
identified with VατHjp+FMan/(τ − z)VατHjp+FMan for each j. Proving that the
later module is strict is a local question, near points with coordinates (τ, z) in the neighbourhood of (τo, zo) with τo= zo.
(1) If τo = zo = 0, we use that VατHjp+FMan/τ VατHjp+FMan is strict, due to
the strict specializability of Hjp
+FManalong τ = 0 and it is enough to prove that z
is injective on Vτ
αHjp+FMan/(τ − z)VατHjp+FMan. Due to the strictness above,
if a local section m of Vτ
αHjp+FMan satisfies zm = τ m′ for some local section m′
of Vτ
αHjp+FMan, then there exists a local section m′′ of VατHjp+FMan such that
m = τ m′′, and since τ is injective on VατHjp+FManfor α ∈ [0, 1), we have m′= zm′′.
As a consequence, if a local section m of Vτ
αHjp+FMan satisfies zm = (τ − z)m1
for some local section m1 of VατHjp+FMan, then there exists a local section m′′ of
Vτ
αHjp+FMansuch that m+ m1= τ m′′and m1= zm′′, hence m = (τ − z)m′′, which
gives the desired injectivity.
(2) We now assume that τo= zo6= 0. Near such a point, we have VατHjp+FMan=
Hjp
+FMan. Let us remark, however, that the V -filtration ofFManalong τ − τo= 0
satisfies V(τ −τo) k FMan=FManfor k > 0 and V(τ −τo) k FMan= (τ −τ o)−kFManfor k 6 0,
according to [Sab06b, Prop. 4.1(iii)]. Applying the push-forward argument as above, we conclude that the V -filtration of Hjp
+FManalong τ − τo = 0 satisfies the same
property. Therefore, setting τ′ = τ − τ
the injectivity of z′on Vτ′
α Hjp+FMan/(τ′− z′)Vτ
′
α Hjp+FMan. We can then use the
same argument as we used for the case τo= 0.
5. The irregular Hodge filtration
In this section, we come back to the setup of Theorem 1.3. Let f be a meromorphic function on X with pole divisor P and let (N, F•N) be a filtered DX-module underlying
a mixed Hodge module such that N = N(∗Pred). Let (M, F•M) be the mixed Hodge
module on X ×P1associated to (N, F
•N) by the construction of Remark 2.3. We know
by Theorem 4.1 that the complex pX,+RFDel • (M ⊗ E
t) is strict and has cohomology
in degree zero at most, hence H0p
X,+RFDel • (M ⊗ E
t) is equal to the Rees module of
N⊗ Ef with respect to some good filtration, which we precisely define as Firr
• (N ⊗ E
f).
Definition 5.1. The filtration Firr
• (N ⊗ E
f) is the filtration obtained by push-forward
from FDel
• (M ⊗ E
t).
5.a. Proof of Theorem 1.3
(1) This is clear since it already holds for FDel
• (M ⊗ E
t).
(2) Because the category of mixed Hodge modules is abelian, we have an exact sequence of filtered D-modules underlying mixed Hodge modules:
0 −→ (N0, F•N0) −→ (N1, F•N1)
ϕ
−−→ (N2, F•N2) −→ (N3, F•N3) −→ 0
which gives rise to an exact sequence of filtered D-modules underlying mixed Hodge modules:
0 −→ (M0, F•M0) −→ (M1, F•M1)
ϕ
−−→ (M2, F•M2) −→ (M3, F•M3) −→ 0
and therefore, according to [ESY15, Th. 3.0.1(2)], to an exact sequence of filtered D-modules:
0 → (M0⊗ Et, F•Del) → (M1⊗ Et, F•Del) → (M2⊗ Et, F•Del) → (M3⊗ Et, F•Del) → 0.
Applying H0p
+ we keep an exact sequence, according to the second statement in
Theorem 4.1.
(3) We consider the following diagram: X × P1 pX π × IdP1 //Y × P1 pY X π //Y We thus have π+RFirr • (N ⊗ E f) ≃ (π ◦ p X)+RFDel • (M ⊗ E t) ≃ (p Y ◦ (π × Id))+RFDel • (M ⊗ E t).
On the other hand, according to [ESY15, Prop. 3.2.3], (π × Id)+RFDel
• (M ⊗ E
t) is
strict and for each j,
Hj(π × Id)+RFDel • (M ⊗ E t) ≃ R FDel • Hj(π × Id)+M⊗ Et. Applying now Theorem 4.1 to Hj(π × Id)
+(M, F•M) we obtain the assertion.
(5) The case 1.1(a) follows from [ESY15, Prop. 1.6.12]. Let us show the case 1.1(b). If i : P1
t ֒→ P1t × P1s denotes the diagonal inclusion t 7→ (t, t) and
p : P1t× P1s→ P1t denotes the projection (and similarly after taking the product
with X), we have an isomorphism
M⊗ Et≃ H0p+(i+(M ⊗ Et)) ≃ H0p+ (i+M) ⊗ Es. We claim that, for each α ∈ [0, 1),
(5.2) Fα+irr•(M ⊗ E t) = FDel α+•(M ⊗ E t). It is enough to check (5.3) i+ M⊗ Et, Fα+Del•(M ⊗ E t)= (i +M) ⊗ Es, Fα+Del•((i+M) ⊗ E s),
and the question is non obvious in the charts t′ = 1/t and s′ = 1/s. Let us set
δ = δ(s′− t′). Let us first recall that, by definition,
i+M′ = L k>0 M′⊗ ∂k s′δ, Fp(i+M′) = L k>0 Fp−k−1M′⊗ ∂ks′δ,
(the shift by one comes from the left-to-right transformation on RFD-modules) and,
concerning the V -filtration, one checks that Vαs′(i+M′) = L k>0 Vα−kt′ M′⊗ ∂sk′δ = X k>0 ∂sk′s ′k(Vt′ αM′⊗ δ).
On the other hand, we have FDel α+p (i+M′) ⊗ E1/s ′ = s′−1 F p(i+M′) ∩ Vs ′ α (i+M′)⊗ e1/s ′ + ∂s′Fα+p−1Del (i+M′) ⊗ E1/s ′ , i+ Fα+Del•(M ′⊗ E1/t′ )p= Fα+p−1Del (M′⊗ E1/t ′ ) ⊗ δ + ∂s′i+ Fα+Del •(M ′⊗ E1/t′ )p−1. We will prove (5.3) by induction on p. Let pobe such that Fpo−2M
′= 0. We have Fα+pDelo (i+M ′) ⊗ E1/s′ = s′−1 Fpo(i+M ′) ∩ Vs′ α (i+M′)⊗ e1/s ′ = s′−1 Fpo−1M ′∩ Vt′ αM′ ⊗ (δ ⊗ e1/s′) = t′−1(Fpo−1M ′∩ Vt′ αM′) ⊗ e1/t ′ ⊗ δ = i+ Fα+Del•(M ′⊗ E1/t′ )p o.
Let us first show by induction on p that i+ Fα+Del•(M
′⊗ E1/t′
)p⊂ Fα+pDel (i+M′) ⊗ E1/s
′
. It is thus enough to check that
FDel α+p−1(M′⊗ E1/t ′ ) ⊗ δ ⊂ FDel α+p (i+M′) ⊗ E1/s ′ . We have Fα+p−1Del (M′⊗ E1/t ′ ) = t′−1(Fp−1M′∩ Vt ′ αM′) ⊗ e1/t ′ + ∂t′ Fα+p−2Del (M′⊗ E1/t ′ ).
Then on the one hand, by induction, ∂t′ Fα+p−2Del (M′⊗ E1/t ′ )⊗ δ ⊂ ∂t′ Fα+p−2Del (M′⊗ E1/t ′ )⊗ δ + ∂s′ Fα+p−2Del (M′⊗ E1/t ′ )⊗ δ = ∂t′Fα+p−1Del (i+M′) ⊗ E1/s ′ + ∂s′Fα+p−1Del (i+M′) ⊗ E1/s ′ ⊂ Fα+pDel (i+M′) ⊗ E1/s ′ . On the other hand, t′−1(F
p−1M′∩ Vt
′
αM′) ⊗ e1/t
′
⊗ δ is the degree zero (w.r.t. δ) term in FDel
α+p (i+M′) ⊗ E1/s
′
.
Let us now prove by induction on p the reverse inclusion Fα+pDel (i+M′) ⊗ E1/s ′ ⊂ i+ Fα+Del•(M ′⊗ E1/t′ )p. It is enough to prove s′−1 Fp(i+M′) ∩ Vs ′ α (i+M′)⊗ e1/s ′ ⊂ i+ Fα+Del•(M ′⊗ E1/t′ )p. The left-hand side reads
X j>0 s′−1 (Fp−j−1M′∩ Vt ′ α−jM′) ⊗ ∂ j s′δ ⊗ e1/s′ =X j>0 s′−1(Fp−j−1M′∩ Vt ′ α−jM′) ⊗ (∂s′+ s′−2)j(δ ⊗ e1/s ′ ) =X j>0 s′−1(∂s′+ s′−2)j (Fp−j−1M′∩ Vt ′ α−jM′) ⊗ e1/t ′ ⊗ δ. Writing ∂s′(m ⊗ δ) = m ⊗ ∂s′δ = −m ⊗ ∂t′δ = (∂t′m) ⊗ δ − ∂t′(m ⊗ δ), we obtain s′−1(∂s′+ s′−2)j (Fp−j−1M′∩ Vt ′ α−jM′) ⊗ e1/t ′ ⊗ δ ⊂ j X i=0 ∂ti′ t′−1∂tj−i′ (Fp−j−1M ′∩ V α−jM′)⊗ e1/t ′ ⊗ δ and since i+ Fα+Del•(M
′⊗ E1/t′
)is an F -filtration, it is enough to check that t′−1∂tj−i′ (Fp−j−1M ′∩ V α−jM′)⊗ e1/t ′ ⊗ δ ⊂ i+ Fα+Del•(M ′⊗ E1/t′ )p−i. Considering the term of degree zero with respect to ∂•
s′δ in the right-hand side, it is
thus enough to check that t′−1∂j−i
t′ (Fp−j−1M
′∩ V
α−jM′) ⊂ t′−1(Fp−i−1M′∩ VαM′).
Now, the assertion is clear.
Remark 5.4. Let f : X → P1be a morphism and let (N, F
•N) underlie a mixed Hodge
module. It follows from 1.3(4) and (5) that if,+(N ⊗ Ef, F•irr) = (M ⊗ E
t, FDel
• ), if we
set as above (M, F•M) = if,+(N, F•N).
5.b. The irregular Hodge filtration in terms of Vτ
α. With the notation as in
the beginning of this section, we consider the pull-back module N[τ ] by the projec-tion r : X × Cτ → X and the corresponding Rees object RFN[τ ] =: N [τ ], where
N := RFN. Denote by Nan the analytification of N and by r+N anthat of N [τ ]. We twist r+N an by Eτ f /z to obtain the object FfN
, which underlies an object of MTMint(X×Cτ), according to Propositions 3.3 and 3.4, and to [Moc11b, Prop. 11.3.4
Proposition 5.5. We have RFirr α (N ⊗ E f)an= Vτ α( FfN )/(τ − z)Vτ α( FfN ).
Proof. We associate to the mixed Hodge module (N, F•N) the mixed Hodge module
(M, F•M) as in Remark 2.3 (from which we keep the notation). It follows from (5.2)
and [ESY15, Prop. 3.1.2] that the result holds for (M, F•M) on X × P
1 and for f
equal to the projection to P1 (note that it holds without taking “an”). Applying
the same argument as in the proof of Theorem 4.1, we conclude that the operation Vτ
α/(τ − z)Vατ commutes with H0p+. On the other hand, by definition and according
to (5.2), the operation RFirr
α is compatible with H
0p
+. Therefore, the result holds
for (N, F•N).
Remark 5.6. As a consequence, one can recover the graded module RFirr α (N ⊗ E
f)
from Vτ α(
FfN
) in the following way. As an OX[z]-module, we have an inclusion
RFirr α (N ⊗ E f) ⊂ N[z, z−1] and R Firr α (N ⊗ E f) is obtained from R Firr α (N ⊗ E f)an as
the graded module with respect to the filtration on RFirr α (N ⊗ E
f)an induced by the
z-adic filtration of OX ⊗OX[z]N[z, z−1]. By the proposition above, it is thus enough
to identify the inclusion as OX-modules
(5.6 ∗) Vτ α( FfN )/(τ − z)Vτ α( FfN ) ⊂ OX ⊗OX[z]N[z, z −1].
By using the strict specializability of FfN along τ = 0, one checks as in [ESY15,
Proof of Prop. 3.1.2] that (τ − z)Vτ α( FfN ) = Vτ α( FfN ) ∩ (τ − z)FfN , so that Vατ( FfN )/(τ − z)Vατ( FfN ) ⊂FfN /(τ − z)FfN . Let us set (recall that N = N(∗Pred))
f N = (RFN)(∗Pred) =L p FpN(∗Pred)zp⊂ N[z, z−1], f N an= OX ⊗OX[z]Nf⊂ OX ⊗OX[z]N[z, z−1]. As an OX×Cτ-module we have FfN = r∗Nfanand thus as an O X-module we have FfN /(τ − z)FfN
= fN an. This gives the desired inclusion (5.6 ∗).
PART II. THE CASE OF A RESCALED MEROMORPHIC FUNCTION
6. Kontsevich bundles via D -modules
In Part II, we use the setting and notation of §1.b. It will also be convenient to work algebraically with respect to P1, in which case we will consider the D
X[v]h∂vi-module
Evf(∗H) := O
X(∗D)[v]·evf and the DX[u]h∂ui-module Ef /u(∗H) := OX(∗D)[u, u−1]·
ef /u, where evf and ef /u are other notations for 1 which make clear the twist of the
connection.
6.a. The Laplace Gauss-Manin bundles Hk(α). The bundles Hk(α) on P1 vwill
Over the chart Cv. Let us denote by Hvk the restriction of Hk (see §1.c) to the
v-chart. This is nothing but the Laplace transform of the (k − dim X)th Gauss-Manin system of f . It is known to have a regular singularity at v = 0 and no other singularity at finite distance (as follows by push-forward from the arguments recalled in §7.b, or by a general result about Laplace transform of regular holonomic D-modules in one variable). Moreover, Hk
v is equipped with the push-forward filtration F•irrH
k v by
OCv-coherent subsheaves, in a strict way according to Theorem 1.3. On C∗v we obtain
in such a way a flat bundle (Hk |C∗
v, ∇) equipped with a filtration F
irr
• H
k |C∗
v indexed
by Q, which satisfies Griffiths transversality condition with respect to ∇ (see §7.f, see also Remark 6.3). This is the variation with respect to v of the irregular Hodge filtration of Hk(X, DR Evf(∗H)).
We consider the limiting filtration (in the sense of Schmid) when v → 0. For α ∈ [0, 1), let us denote by VαHvk the αth term of the Kashiwara-Malgrange
filtration of Hk
v at v = 0. Equivalently, due to the regularity property of the
connection at v = 0, VαHvk is the Deligne extension of H|Ck∗
v on which ∇ has a
simple pole with residue Resv=0∇ having eigenvalues in [−α, −α + 1). We set
grV
αHvk = VαHvk/V<αHvk. Theorem 6.1. For each α ∈ [0, 1),
(1) the jumps β ∈ Q of the induced filtration F•irrgrVαHvk:= Firr • H k∩ V αHvk Firr • H k∩ V <αHvk belong to α + Z,
(2) on each generalized eigenspace of Resv=0∇ acting on VαHvk/vVαHvk, the
nilpotent part of the residue strictly shifts by one the filtration naturally induced by
Firr
• VαH
k v .
Our proof in §7.i is obtained by showing (Proposition 7.19) that the conditions needed for applying M. Saito’s criterion [Sai88, Prop. 3.3.17] are fulfilled. More pre-cisely, we will work with a filtration F•E
vf(∗H) easy to define, and we postpone to §9
the proof that this is indeed the irregular Hodge filtration of Evf(∗H). It would
also be possible, as observed by T. Mochizuki [Moc15], to directly refer to a similar property for twistor D-modules.
Over the chart Cu. Let us now consider the chart Cu. We denote by Huk the
restric-tion of Hk to this chart. If j : C
ur{0} ֒→ Cu denotes the open inclusion, we have
Hk
u = j+H|Ck∗
v. There is a natural OCu-lattice G0
Hk
u of the free OCu[u
−1]-module
Hk
u called the Brieskorn lattice in analogy with the construction of Brieskorn in
singu-larity theory [Bri70]. It can be defined in terms of the Hodge filtration of the Gauss-Manin system attached to f (see the appendix). It can also be defined (see §8.d) as the push-forward by q in a suitable sense of an OX×Cu(∗Pred)-module G0E
f /u(∗H)
equipped with a u-connection ud + df : G0Ef /u(∗H) → G0Ef /u(∗H) ⊗ Ω1X and with
a compatible action of u2∂u.
The connection on Hk has a pole of order at most two at u = 0 when restricted
to G0Huk (see Remark 8.14). In the context of DX[u]h∂ui-modules Ef /u(∗H)
corre-sponds to Ef /u(∗H) = O
X(∗D)[u, u−1]ef /u.
Gluing. We can then glue G0Huk with VαHvk and obtain an OP1-bundle Hk(α) with
6.b. The Kontsevich bundles Kk(α). We now consider the Kontsevich bundles introduced in §1.c. We can endow them with a natural meromorphic connection having a pole of order one at v = 0 and of order two at most at v = ∞, and no other pole.
In order to do so, we start(2) by considering the morphism of complexes
u2∂u− f : Ω•f(α)[u], ud + df −→ Ω• f(α + 1)[u], ud + df .
Lemma 6.2. For α ∈ [0, 1), the natural inclusion of complexes
(Ω•
f(α)[u], ud + df ) −→ (Ω•f(α + 1)[u], ud + df )
is a quasi-isomorphism.
This lemma allows us to define an action of u2∂
u on each Kk(α)|Cu and therefore
a meromorphic connection ∇ on Kk(α) with a pole of order at most two at u = 0.
We will show that ∇ has at most a simple pole at v = 0.
Remark 6.3 (due to T. Mochizuki). Let H be a vector bundle on P1 equipped with
a connection ∇ having a simple pole at v = 0 and a double pole (at most) at v = ∞. Then the Harder-Narasimhan filtration F•H satisfies the Griffiths
transver-sality property with respect to ∇.
Indeed, the property is obviously true with respect to the connection d on H coming from d on each summand in a Birkhoff-Grothendieck decomposition. We are thus reduced to proving a similar property for the O-linear morphism ∇ − d, and the result follows by noticing that (H /Fp−1H)⊗Ω1
P1({v = 0}+2{u = 0}) has slopes < p
while FpH has slopes > p.
Proof of Lemma 6.2. We will show that the quotient complex has zero cohomology.
From [ESY15, Prop. 1.4.2] we know that the inclusion of complexes (Ω•
f(α), df ) −→ (Ω
•
f(α + 1), df )
is a quasi-isomorphism, and thus the quotient complex (Q•, df ) has zero cohomology.
Let ω = Pkj=0ωjuj be a local section of Qp[u] such that (ud + df )(ω) = 0. Then
df ∧ω0= 0 and therefore there exists η0∈ Qp−1such that ω = df ∧η0. By replacing ω
with ω − (ud + df )η0 and iterating the process we can assume that ω = ωkuk and,
dividing by uk, that ω ∈ Qp. It satisfies then dω = 0 and df ∧ω = 0, so ω = df ∧η for
some η ∈ Qp−1, and therefore df ∧ dη = 0. For any representativeη ∈ Ωe p−1
f (α + 1),
we obtain
df ∧ dη ∈ Ωe p+1f (α) ⊂ Ωp+1X (log D)([αP ]). On the other hand, we note that
Ωp−1f (α + 1) = df ∧ Ωp−2X (log D)([αP ]) + Ωp−1X (log D)([αP ]), so we can assume that eη ∈ Ωp−1X (log D)([αP ]). Then dη ∈ Ωe
p
X(log D)([αP ]), and
therefore dη ∈ Ωe pf(α), that is, dη = 0, so ω = (ud + df )η.
Proof of Theorem 1.11. We will compare the filtered complex σ>p(Ω•
f(α)[v], d + vdf )
with the filtered relative de Rham complex of Evf(∗H) with respect to the projection
to Cv. We introduce in §7.c a filtration Fα•Evf(∗H), which will be shown to coincide
with Firr,•
α Evf(∗H) in Theorem 9.1.
Theorem 6.4 (see §7.g, modulo Th. 9.1). There is a natural quasi-isomorphism of filtered complexes (OX×Cv ⊗OX[v]Ω • f(α)[v], d + vdf, σ>p) −→ (DRX×Cv/CvVαE vf(∗H), Firr,p α )
which is compatible with the meromorphic action of ∇∂v.
It follows from (1.9) that applying Rq∗ to the filtered complex on the right-hand
side gives a strict complex (i.e., we have a similar injectivity statement). We apply Rkq
∗ to the quasi-isomorphism of Theorem 6.4. The non-filtered
state-ment gives the first point of Theorem 1.11, since Vαis compatible with proper
push-forward. The second point is then obtained by applying the second point of Theo-rem 6.1.
In a way similar to Theorem 6.4, but algebraically with respect to u, we in-troduce in §8.b a filtration F•
αG0Ef /u(∗H), which will be shown to coincide with
Firr,•
α G0Ef /u(∗H) in Theorem 9.1, and we prove:
Theorem 6.5 (see §8.c, modulo Th. 9.1). There is a natural quasi-isomorphism of filtered complexes
(Ω•
f(α)[u], ud + df, σ>p) −→ (DRXG0Ef /u(∗H), Fαirr,p)
which is compatible with the action of ∇∂u.
As above, it follows from (1.9) that applying Rq∗ to the filtered complex on the
right-hand side gives a strict complex.
By applying a degeneration statement similar to that of [Sai88, Prop. 3.3.17] proved in the appendix, we obtain a concrete description of the irregular Hodge fil-tration of Hk.
Corollary 6.6. The isomorphism Kk(α)−→ H∼ k(α) obtained by pushing forward the
quasi-isomorphisms of Theorems 6.4 and 6.5 identifies the Harder-Narasimhan
filtra-tion of Kk(α) (hence of Hk(α)) with the image on Hk(α) of the irregular Hodge
filtration FirrHk.
Remark 6.7. Another proof of Theorem 1.11 has recently been given by T. Mochizuki [Moc15], by showing an analogue of Theorem 6.4 in the framework of mixed twistor D-modules, but not referring explicitly to the irregular Hodge filtration.
7. The DX×Cv-module E
vf(∗H)
7.a. Setting. We will use the local setting and notation similar to that of [ESY15, §1.1] that we recall now, together with the notation introduced in §1.b. In the local analytic setting, the space Xan is the product of discs ∆ℓ× ∆m× ∆m′
with coor-dinates (x, y) = (x1, . . . , xℓ, y1, . . . , ym, y1′, . . . , ym′ ′) and we are given a multi-integer
